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I don't know if this is true of conformal mapping or just mapping in general but I want to be completely sure that if I know how the contour of a region transforms then the points within the original contour will be inside the transformed contour as well.

If this isn't clear consider the opposite: I know how the contour transforms and it is a unit circle centered at the origin. Could there be a point within the original region that lies outside this contour?

Xander Henderson
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DLV
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Holomorphic functions are open maps; that is, if you can find an open set around it in the original region (i.e., it isn't a part of the boundary), then whatever point it gets mapped to will have a corresponding open set around it. Therefore, it can't be an element of the boundary of the image.

Clayton
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  • I'm sorry but I'm not familiar with this terminology, I'll have to look it up. If you can provide a more simple answer I'll be glad to accept, but in the meanwhile I'll read up on these terms. Specifically I don't know what open set is, hehe. – DLV Sep 14 '15 at 02:02
  • Is what you say easy to prove or is it non-trivial? – DLV Sep 14 '15 at 02:07